We obtain a qualitative extension of Selberg’s 3/16 Theorem to the setting of moduli spaces of abelian differentials on genus g≥2 surfaces. More precisely, under certain conditions we prove that there is a uniform spectral gap for the foliated hyperbolic Laplacian operators associated to congruence covers of a fixed component of a stratum of the moduli space of abelian differentials on a genus g surface. We also state our result in representation-theoretic terms. To prove our result, we build on the work of Avila, Gouëzel and Yoccoz (2006) on exponential mixing of the Teichmüller flow as well as the group expansion technology developed by Bourgain, Gamburd and Sarnak (2011), Oh and Winter (2014) and Bourgain, Kontorovich and Magee (2015). Our results apply to hyperelliptic components and also extend to arbitrary components on assumption of a conjecture of Zorich that asserts the Zariski-density of the associated Rauzy-Veech group in its ambient symplectic group.